GENERAL ⎜ ARTICLE
Brownian Motion Problem: Random Walk and
Beyond
Shama Sharma and Vishwamittar
A brief account of developments in the experimental and theoretical investigations of Brownian motion is presented. Interestingly, Einstein
who did not like God’s game of playing dice for
electrons in an atom himself put forward a theory
of Brownian movement allowing God to play the
dice. The vital role played by his random walk
model in the evolution of non-equilibrium statistical mechanics and multitude of its applications
is highlighted. Also included are the basics of
Langevin’s theory for Brownian motion.
Introduction
Brownian motion is the temperature-dependent perpetual, irregular motion of the particles (of linear dimension of the order of 10¡6m) immersed in a fluid, caused
by their continuous bombardment by the surrounding
molecules of much smaller size (Figure 1). This e®ect
was first reported by the Dutch physician, chemist and
engineer Jan Ingenhousz (1785), when he found that
fine powder of charcoal floating on alcohol surface exhibited a highly random motion. However, it got the
name Brownian motion after Scottish botanist Robert
Brown, who in 1828-29 published the results of his extensive studies on the incessant random movement of
tiny particles like pollen grains, dust and soot suspended
in a fluid (Box 1). To begin with when experiments
were performed with pollen immersed in water, it was
thought that ubiquitous irregular motion was due to the
life of these grains. But this idea was soon discarded as
the observations remained unchanged even when pollen
was subjected to various killing treatments, liquids other
RESONANCE ⎜ August 2005
Shama Sharma is
currently working as
lecturer at DAV College,
Punjab. She is working on
some problems in
nonequilibrium statistical
mechanics.
Vishwamittar is professor
in physics at Panjab
University, Chandigarh
and his present research
activities are in
nonequilibrium statistical
mechanics. He has written
articles on teaching of
physics, history and
philosophy of physics.
Keywords
Bro w ni a n m otion, L a n g evin’s
th e ory, r a ndom w a lk, stoch a stic processes, non- e q uilibriu m
st a tistica l m ech a nics.
49
GENERAL ⎜ ARTICLE
Figure 1. Brownian motion
of a fine particle immersed
in a fluid.
than water were used and fine particles of glass, petrified wood, minerals, sand, etc were immersed in various
liquids. Furthermore, possible causes like vibrations of
building or the laboratory table, convection currents in
the fluid, coagulation of particles, capillary forces, internal circulation due to uneven evaporation, currents
produced by illuminating light, etc were considered but
were found to be untenable.
An early realization of the consequences of the studies on Brownian motion was that it imposes limit on
the precision of measurements by small instruments as
these are under the influence of random impacts of the
surrounding air.
Box 1. Observing the Brownian Motion
One can observe Brownian movement by looking through an optical microscope with magnification 103
– 104 at a well illuminated sample of some colloidal solution (say milk drops put into water) or smoke
particles in air. Alternately, we can mount a small mirror (area 1 or 2 mm2) on a fine torsion fiber capable
of rotation about a vertical axis (as in suspension type galvanometer) in a chamber having air pressure of
about 10–2 torr and note the movement of the spot of light reflected from the mirror. The trace of the light
spot is manifestation of angular Brownian motion of the mirror.
It may be pointed out that movement of dust particles in the air as seen in a beam of light through a window
is not an example of Brownian motion as these particles are too large and the random collisions with air
molecules are neither much imbalanced nor strong enough to cause Brownian movement.
50
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GENERAL ⎜ ARTICLE
In 1863, Wiener attributed Brownian motion to molecular movement of the liquid and this viewpoint was
supported by Delsaux (1877-80) and Gouy (1888-95).
On the basis of a series of experiments Gouy convincingly ruled out the exterior factors as causes of Brownian motion and argued in favour of contribution of the
surrounding fluid. He also discussed the connection between Brownian motion and Carnot’s principle and thereby brought out the statistical nature of the laws of thermodynamics. Such ideas put Brownian motion at the
heart of the then prevalent controversial views about
philosophy of science. Then, in 1900 Bachelier obtained
a di®usion equation for random processes and thus, a
theory of Brownian motion in his PhD thesis on stock
market fluctuation. Unfortunately, this work was not
recognized by the scientific community, including his
supervisor Poincaré, because it was in the field of economics and it did not involve any of the relevant physical
aspects.
Eventually, Einstein in a number of research publications beginning in 1905 put forward an acceptable model
for Brownian motion1 . His approach is known as ‘random walk’ or ‘drunkards walk’ formalism and uses the
fluctuations in molecular collisions as the cause of Brownian movement. His work was followed by an almost similar expression for the time dependence of displacement
of the Brownian particle by Smoluchowski in 1906, who
started with a probability function for describing the
motion of the particle. In 1908, Langevin gave a phenomenological theory of Brownian motion and obtained
essentially the same formulae for displacement of the
particle. Their results were experimentally verified by
Perrin (1908 and onwards)2 using precise measurements
on sedimentation in colloidal suspensions to determine
the Avogadro’s number. Later on, more accurate values of this constant and that of Boltzmann constant k
were found out by many workers by performing similar
experiments.
RESONANCE ⎜ August 2005
In 1863, Wiener
attributed Brownian
motion to molecular
movement of the
liquid.
1
These are collected in the book
A l b ert Einst e in : Inv esti g a tions
o n t h e t h e o ry o f Bro w n i a n
Move m e nt edited with notes by
R Furth (Dov e r Pu b lic a tio ns,
Ne w York,1956).
2
Revie w article entitle d ‘Browni a n M ov e m e nt a n d M ol ecul a r
Re a lity’ b a se d on his w ork h a s
b e e n tr a nsl a t e d into En glish by
F So d dy a n d p u b lish e d by T a ylor & Fr a ncis,London in 1910.
51
GENERAL ⎜ ARTICLE
The correctness of
the random walk
model and
Langevin’s theory
made a very strong
case in favour of
molecular kinetic
model of matter.
The techniques
developed for the
theory of Brownian
motion form
cornerstones for
investigating a variety
of phenomena.
52
The correctness of the random walk model and Langevin’s theory made a very strong case in favour of molecular kinetic model of matter and unleashed a wave of
activity for a systematic development of dynamical theory of Brownian motion by Fokker, Planck, Uhlenbeck,
Ornstein and several other scientists. As such, during
the last 100 years, concerted e®orts by a galaxy of physicists, mathematicians, chemists, etc have not only provided a proper mathematical foundation to the physical
theory but have also led to extensive diverse applications
of the techniques developed.
The two basic ingredients of Einstein’s theory were: (i)
the movement of the Brownian particle is a consequence
of continuous impacts of the randomly moving surrounding molecules of the fluid (the ‘noise’); (ii) these impacts
can be described only probabilistically so that time evolution of the particle under observation is also probabilistic in nature. Phenomena of this type are referred to
as stochastic processes in mathematics and these are essentially non-equilibrium or irreversible processes. Thus,
Einstein’s theory laid the foundation of stochastic modeling of natural phenomena and formed the basis of
development of non-equilibrium statistical mechanics,
wherein generally, the Brownian particle is replaced by
a collective property of a macroscopic system. Consequently, the techniques developed for the theory of
Brownian motion form cornerstones for investigating a
variety of phenomena (in di®erent branches of science
and engineering) that have their origin in the e®ect of
numerous unpredictable and may be unobservable events
whose individual contribution to the observed feature is
negligible, but collective impact is observed in the form
of rather rapidly varying stochastic forces and damping
e®ect (see Box 2).
The main mathematical aspect of the early theories of
Brownian motion was the ‘central limit theorem’, which
am- ounted to the assumption that for a su±ciently large
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GENERAL ⎜ ARTICLE
Box 2. Brownian Motors
One of the topics of current research activity employing these techniques is referred to as Brownian motors.
This phrase is used for the systems rectifying the inescapable thermal noise to produce unidirectional
current of particles in the presence of asymmetric potentials. Their study is relevant for understanding the
physical aspects involved in the movement of motor proteins, the design and construction of efficient
microscopic motors, development of separation techniques for particles of micro- and nano-size, possible
directed transport of quantum entities like electrons or spin degrees of freedom in quantum dot arrays,
molecular wires and nano-materials; and to the fundamental problems of thermodynamics and statistical
mechanics such as basing the second law of thermodynamics on statistical reasoning and depriving the
Maxwell demon type devices of their mystique and the trade-off between entropy and information.
number of steps, the distribution function for random
walk is quite close to a Gaussian. However, since the
trajectory of a Brownian particle is random, it grows
only as square root of time3 and, therefore, one cannot
define its derivative at a point. To handle this problem,
N Wiener (1923) put forward a measure theory which
formed the basis of the so-called stable distributions or
Levy distributions. As a follow-up of these and to give
a firm footing to the theory of Brownian motion, Ito
(1944), developed stochastic calculus and an alternative
to Brownian motion — the Geometrical Brownian motion. This, in turn, led to extensive modeling for the
financial market, where the balance of supply and demand introduces a random character in the macroscopic
price evolution. Here the logarithm of the asset price is
governed by the rules for Brownian motion. These aspects have enlarged the scope of applicability of theoretical methods far beyond Einstein’s random walks. These
concepts have been fruitfully exploited in a multitude of
phenomena in not only physics, chemistry and biology
but also physiology, economics, sociology and politics.
The Random Walk Model for Brownian Motion
While tottering along, a drunkard is not sure of his steps;
these may be forward or backward or in any other direction. Besides, each step is independent of the one
already
RESONANCE ⎜ August 2005
3
This result w ill b e o bt a in e d in
th e n e xt s e ctio n of this a rticl e ;
se e e q u a tion (8).
The main
mathematical aspect
of the early theories
of Brownian motion
was the ‘central limit
theorem’, which
states that the
distribution function
for random walk is
quite close to a
Gaussian.
53
GENERAL ⎜ ARTICLE
taken. Thus, his motion is so irregular that nothing
can be predicted about the next step. All one can talk
about is the probability of his covering a specific distance
in given time. Such a problem was originally solved
by Markov and therefore, such processes are known as
Markov processes. Einstein adopted this approach to
obtain the probability of the Brownian particle covering
a particular distance in time t. The randomness of the
drunkard’s walk has given this treatment the name ‘random walk model’ and in the case of a Brownian particle,
the steps of the walk are caused by molecular collisions.
To simplify the derivation, we consider the problem in
one-dimension along x-axis, taking the position of the
particle to be 0 at t = 0 and x(t) at time t and make
the following assumptions.
1. Each molecular impact on the particle under observation takes place after the same interval ¿0(10¡8 s) so
that the number of collisions in time t is n = t/¿0 (of
the order of 108).
2. Each collision makes the Brownian particle jump by
the same distance ± which turns out, we will see, to be
about 1nm, for a particle of radius 1µm in a fluid with
the viscosity of water, at room temperature along either
positive or negative direction with equal probability and
± is much smaller than the displacement x(t) which we
resolve through the microscope only on the scale of a
µm of the particle.
3. Successive jumps of the particle are independent of
each other.
4. At time t, the particle has net positive displacement
x(t) (it could be equally well taken as negative) so that
out of n jumps, m(= x(t)/±, of the order of 103) extra
jumps have taken place in positive direction. Since n
and m are quite large, these are taken as integers.
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GENERAL ⎜ ARTICLE
In view of the four assumptions above, the total number
of positive and negative jumps, respectively, is (n+m)/2
and (n ¡ m)/2. Since the probability that n independent jumps, each with probability 1/2, have a particular sequence is (1/2)n , the probability of having m extra
positive jumps is given by
P n(m) = (1/2)(1/2)n{n!/[(n + m)/2]![(n ¡ m)/2]!},
(1)
where the factor (1/2) comes from normalization § m
Pn (m) = 1. While writing this expression, we have not
used any restriction on n and m except that both are
integers. So it holds good even for small n and m. Note
that both n and m are either even or odd. As an illustration, if we consider 8 jumps, then the probabilities
for m = 0, 2, 4 and 8 are 35/256, 7/64, 7/128 and 1/512,
respectively.
For large n we can use the approximation n! = (2¼n)1/2
(n/e)n, so that for large n and m, equation(1) finally
becomes
Pn (m) = (1/2n)1/2 exp(¡m2/2n).
(2)
Substituting m = x/± and n = t/¿0, this yields the
probability for the particle to be found in the interval x
to x + dx at time t as
P (x)dx = (1/4¼Dt)1/2exp( ¡x2/4Dt)dx,
(3)
with D = ± 2/2¿0 . It may be mentioned that taking m to
be large mathematically means that ± is infinitesimally
small for a finite value of x. This aspect has enabled
us to switch over from discrete step random walk to a
continuous variable x in the above expression.
In order to understand the physical significance of the
symbol D, recall that if the linear number density of
suspended particles in a fluid is N(x, t) and their concentration is di®erent in di®erent regions, di®usion takes
RESONANCE ⎜ August 2005
55
GENERAL ⎜ ARTICLE
place. This is governed by the di®usion equation
D0(@ 2 N(x, t)/@x2) = @N(x, t)/@t,
(4)
where D0 is the di®usion coe±cient. Equation (4) has
one of the solutions as
p
N(x, t) = (Ntot / {4¼D0 t})exp(¡x2 /4D0 t).
(5)
Here,
Ntot =
Z
1
N(x, t)dx
(6)
¡1
is the total number of particles along the x-axis. Obviously, N (x, t)/Ntot corresponds to P (x) if we identify
D as D0 , i.e, ±2 /2¿0 as di®usion coe±cient. In other
words, P (x) is a solution of the di®usion equation meaning thereby that the random walk problem is intimately
related with the phenomenon of di®usion. In fact, the
random walk model is a microscopic phenomenon of diffusion.
A look at (3) shows that the probability function describing the position of the particle at time t is Gaussian
with dispersion 2Dt (Figure 2) implying spread of the
peak width as square root of t. Furthermore, mean values of x(t) and x2(t) turn out to be
hx(t)i = 0
The proportionality of
root mean square
displacement of the
Brownian particle to
the square root of
time is a typical
consequence of the
random nature of the
process.
56
and
hx2(t)i = 2Dt.
(7)
Since hx(t)i = 0, the physical quantity of interest is
hx2 (t)i . So what one measures experimentally is the
root mean square displacement of the Brownian particle
given by
xrms = hx2(t)i 1/2 = (2Dt)1/2.
(8)
The proportionality of root mean square displacement
of the Brownian particle to the square root of time is a
typical consequence of the random nature of the process
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GENERAL ⎜ ARTICLE
1.2
1
0.8
Series1
Series2
Series3
0.6
0.4
0.2
and is in contrast with the result of usual mechanical
predictable and reversible motion where displacement
varies as t. For three dimensional motion, the above
expression becomes
hr2(t)i = hx2 (t)i + hy2(t)i + hz 2(t)i = 6Dt.
(9)
From equation (7), we have
D = h x2(t)i/2t,
4.86
4.57
4.28
3.7
3.99
3.41
3.12
2.83
2.54
2.25
1.96
1.67
1.38
0.8
1.09
0.51
-0.1
0.22
-0.4
-0.7
-0.9
-1.2
-1.5
-1.8
-2.1
-2.4
-3
-2.7
-3.3
-3.6
-3.8
-4.1
-4.4
-5
-4.7
0
Figure 2. Plot of (4p Dt)1/2
P(x) (equation(3)) versus x
for D= 5.22 × 10–13 m2 s–1 and
x varying from 5.0 mm to
5.0 mm corresponding to t
values 1s (the innermost
curve), 2s (the middle
curve), and 4s (the outermost curve ).
(10)
which connects the macroscopic quantity di®usion constant D or D0 with the microscopic quantity the mean
square displacement due to jumps. It gives us a direct relationship between the di®usion process, which is
irreversible, and Brownian movement that has its origin in random collisions. In other words, irreversibility
has something to do with the random fluctuating forces
acting on the Brownian particle due to impacts of the
molecules of the surrounding fluid.
RESONANCE ⎜ August 2005
57
GENERAL ⎜ ARTICLE
4
O n e such w e ll-inv e sti g a t e d
to p ic is th a t of q u a ntu m r a nd o m w a lks – a n a tur a l e xt e nsion of cl assica l ra ndom w a lks.
Thou g h th e m a in e m p h a sis in
th e s e stu d i e s h a s b e e n o n
brin g in g out th e d iff e r e nc e s in
th e ir b e h a viour a s com p a re d
to th e cl a ssic a l a n a lo gs, it is
b e li e v e d th a t this kn o w l e d g e
will b e h elpful in d eveloping
a lgorithms th a t ca n b e run on a
q u a n t u m co m p u t e r a s a n d
w h e n it is pr a ctic a lly re a liz e d.
The successful use of the concept of random walk in the
theory of Brownian motion encouraged physicists to exploit it in di®erent fields involving stochastic processes.4
Langevin’s Theory of Brownian Motion
The intimate relationship between irreversibility and randomness of collisions of the fluid molecules with the
Brownian particle (mentioned above) prompted Langevin
(1908) to put forward a more logical theory of Brownian movement. However, before proceeding further, we
digress to consider the motion of a hockey ball of mass
M being hit quickly by di®erent players. Its motion is
governed by two factors. The frictional force f d between
the surface of ball and the ground tends to slow it down,
while the impact of the hit with the stick by a player increases its velocity. The e®ect of the latter, of course, is
quite random as its magnitude as well as direction depend upon the impulse of the hit. Since a hit lasts for
only a very short time, we represent the magnitude fj of
corresponding force at time tj by a Dirac delta function
( Box 3 ) of strength Cj
fj (t) = Cj ±(t ¡ tj ),
(11)
and write the equation of motion pertaining to this force
as
M |dv/dt| = fj .
(12)
To find the e®ect of the hit in changing the velocity,
we substitute (11) into (12), integrate on both the sides
with respect to time over a small time interval around
tj and use the property of delta function to get
M | ¢vj | = C j.
(13)
Here, ¢vj is the change in velocity brought about by
the hit at time t = tj. Now, the hits applied by di®erent
players can be in any direction, the total force exerted
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GENERAL ⎜ ARTICLE
Box 3. The Dirac Delta Function
Many times we come across problems in which the sources or causes of the observed effects are nearly
localized or almost instantaneous. Some such examples are : point charges and dipoles in electrostatics;
impulsive forces in dynamics and acoustics; peaked voltages or currents in switching processes; nuclear
interactions, etc. To handle such situations, we use what is known as Dirac delta function. It represents an
infinitely sharply peaked function given, for one-dimensional case, by
⎧ 0
δ ( x − x 0 ) = ⎨
⎩∞
x ≠ x0
x = x0
but such that its integral over the whole range is normalized to unity :
∞
∫ δ ( x − x 0 )dx = 1.
−∞
Its most important property is that for a continuous function f(x)
∞
∫ f (x )δ ( x − x 0 ) dx = f ( x0 ).
−∞
Thus, δ (x – x0) acts as a sieve that selects from all possible values of f(x) its value at the point x0, where
the δ function is peaked. Accordingly, the above result is sometimes called sifting property ofδ function.
on the ball in a finite time interval will be obtained by
summing up the above expression over a sequence of hits
made taking into account the directions of the hits. This
force can be written as
fr (t) = §j Cj Áj ±(t ¡ tj ),
(14)
where Á j is unit vector in the direction of hit at time
t = tj . The subscript r in (14) has been used to indicate
the randomness of the force vector. Thus, the total force
acting on the ball at any time t will be f d + fr and its
equation of motion will read
M dv/dt = f d + f r .
(15)
Furthermore, if a very large number of hits are applied
quickly over a long time interval as compared to the
impact time, all possible directions Áj will occur completely randomly with equal probability. Accordingly,
RESONANCE ⎜ August 2005
59
GENERAL ⎜ ARTICLE
Figure 3. A large Brownian
particle surrounded by randomly moving relatively
smaller molecules of the
fluid.
average value of fr will be zero, i.e.,
hf r (t)i = 0.
Reverting to the Langevin’s theory, we consider Brownian particle in place of the hockey ball in the above illustration and make the following simplifying assumptions.
1. The Brownian particle experiences a time-dependent
force f(t) only due to molecular impacts from the surrounding particles (Figure 3) and there is no other external force ( such as gravitational, Coulombic, etc.) acting
on it.
2. The force f(t) is made up of two parts: f d and f r (t).
The former is an averaged out, systematic or steady
force due to frictional or viscous drag of the fluid, while
the latter is a highly fluctuating force or noise arising
from the irregular continuous bombardment by the fluid
molecules. The equation of motion of the particle is
given by (15).
3. The steady force fd is governed by the Stokes formula
(Box 4) so that
fd = ¡°v.
60
(16)
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GENERAL ⎜ ARTICLE
Box 4. Stokes Formula
Following Stokes we assume that a perfectly rigid body (say, a sphere) with completely smooth surface and
linear dimension l is moving through an incompressible, viscous fluid ( viscosity η, density ρ ) having an
infinite extent in such a manner that there is no slip between the body and the layers of the fluid in its
contact, with such a speed v that the motion is streamlined and the Reynolds number R ( = vlρ/η) is very
small (strictly speaking vanishingly small), then the viscous drag force fd on the body is proportional to
velocity v, i.e. fd = – γ v; and the constant γ depends on the geometry of the body. For example, it is 6πηa
for a sphere of radius a and is 16 ηa for a plane circular disc of radius a moving perpendicular to its plane.
Here, ° is viscous drag coe±cient and
1/° = |v|/|fd |
(17)
is mobility of the Brownian particle. The drag coe±cient
depends upon viscosity ´ of the fluid, geometry as well
as size of the Brownian particle. For a spherical particle
of radius a,
° = 6¼´a.
(18)
4. The coe±cient of viscous drag ° is a constant, having
the same value in all parts of the fluid and at all times.
5. The random force fr (t) is independent of v(t) and
varies extremely fast as compared to it so that over a
long time interval (much larger than the characteristic
relaxation time ¿), the average of f r (t) vanishes (as elaborated above for a hockey ball), i.e.,
hfr (t)i = 0.
(19)
This essentially amounts to saying that the collisions
are completely independent of each other or are uncorrelated.
Substituting (16) into (15), we get
M (dv/dt) = ¡ °v + fr (t),
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(20)
61
GENERAL ⎜ ARTICLE
where fr (t) satisfies the condition given in (19).This is
the celebrated Langevin equation for a free Brownian
particle. fr (t) is usually referred to as ‘stochastic force’;
(20) is the stochastic equation of motion and is the basic
equation used for describing a stochastic process. The
solution to (20) reads
Z t
¡
¡
v(t) = v(0)exp( t/¿ ) + exp( t/¿ )
exp(u/¿ )A(u)du,
0
(21)
and gives us the drift velocity of the Brownian particle.
Here,
¿ = M/°
(22)
is the relaxation time of the particle and
A(t) = fr (t)/M
(23)
is the stochastic force per unit mass of the Brownian
particle and is generally called ‘Langevin force’.
It may be mentioned that the first term in (21) is the
damping term and it tries to exponentially reduce the
value of v(t) to make it essentially dead. On the other
hand, the second term involving Langevin force or noise
creates a tendency for v(t) to spread out over a continually increasing range of values due to integration and,
thus, keeping it alive. Consequently, the observed value
of drift velocity at any time t is an outcome of these two
opposing tendencies.
From (19) and (23), it is clear that hA(t)i = 0 so that
the average value of drift velocity in (21) becomes
hv(t)i = v(0)exp(¡ t/¿ ).
(24)
Obviously, average drift velocity of the Brownian particle decays exponentially with a characteristic time ¿ till
it becomes zero. This is a typical result for dissipative
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GENERAL ⎜ ARTICLE
or irreversible processes. But, this is physically not acceptable as ultimately the particle must be in thermal
equilibrium with its surroundings at ambient temperature and, therefore, cannot be at rest. So we consider
h v2(t)i = hv(t).v(t) i. Substituting for v(t) from (21)
and using the fact that hA(t)i = 0, we have
hv2(t)i = v2(0)exp(¡2t/¿) + exp(¡2t/¿ )£
Z tZ
0
t
0
exp(u1 + u2)/¿ )K(u2 ¡ u1)du1du2,
(25)
where
K (u2 ¡ u1) = hA(u1).A(u2)i.
(26)
It is called autocorrelation function for the Langevin
force A(u) and involves u2 ¡ u1 to emphasize the fact
that for Brownian motion it depends upon the time interval u2 ¡ u1 rather than actual values of u1 and u2.
We assume that K(u2 ¡ u1) is an even function of its
argument and is significant only for u2 almost equal to
u1. Using these ideas when the double integral in (25)
is evaluated, we get5
2
2
2
2
hv (t)i = v (0) + [v (1) ¡ v (0)][1 ¡ exp(¡2t/¼)].
(27)
5
For d e t a ils of this ste p se e
S t a tistic a l M ech a nics b y R K
P a thri a list e d a t th e e n d .
Here, hv2( 1)i is the mean square velocity of the Brownian particle for infinitely large value of t and is expected
to be the value corresponding to the situation when the
system is in equilibrium at temperature T . Thus, from
equipartition theorem
(1/2)M v2 (1) = (3/2)kT.
(28)
As a special case if v2 (0) equals the equipartition value
3kT /M, then hv2(t)i = v2(0) = 3kT /M implying that
if the system has attained statistical equilibrium then it
would continue to be so throughout.
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63
GENERAL ⎜ ARTICLE
It may be noted that multiplying (20) with r(t)/M,
where r(t) is the instantaneous position of the Brownian
particle, we finally get
d2hr 2i/dt2 + (1/¿ )(dhr 2i/dt) = 2h v2(t)i;
(29)
here h v2(t)i is given by (27). This second order di®erential equation, on being solved, gives
hr2 (t)i = v2(0)¿ 2{1 ¡ exp(¡ t/¿ )}2 ¡ (3kT /M )¿ 2 £
{1 ¡ exp(¡t/¿)}{1 ¡ exp(¡ t/¿ )} + (6kT ¿ /M)t. (30)
For t very small as compared to ¿ it yields
hr2 (t) i = v2(0)t2,
(31)
so that root mean square displacement rrms = hr2(t)i1/2
is proportional to t as for a reversible process. On the
other hand, for t À ¿ , we get
hr 2(t)i = (6kT ¿ /m)t = (6kT /°)t
The Einstein relation
gives a relationship
between the diffusion
coefficient and
mobility of the particle
and hence the
viscosity of the fluid.
(32)
implying that
rrms (t) = (6kT /°)1/2t1/2 .
(33)
Expression (32) becomes identical to (9) if we take
D = kT /°,
(34)
which in view of (18) reads
Thus, viscosity of a
medium is a
consequence of
fluctuating forces
arising from their
continuous and
random motion and is
an irreversible
phenomenon.
64
D = kT /6¼´a
(35)
for a spherical Brownian particle. Equation (34) is usually known as ‘Einstein relation’. Obviously, it gives a
relationship between the di®usion coe±cient and mobility of the particle and hence the viscosity of the fluid.
Thus, viscosity of a medium is a consequence of fluctuating forces arising from their continuous and random
motion and is an irreversible phenomenon.
RESONANCE ⎜ August 2005
GENERAL ⎜ ARTICLE
At this stage it is in order to mention one of the experimental observations by Perrin and Chaudesaigues.
They studied the mean square displacement of gamboge
particles of radius 0.212 µm in a medium with viscosity 0.0012 Nm¡2 s and maintained at a temperature of
186 K. The values found along the x-axis at times 30,
60, 90 and 120 seconds, respectively, were 45, 86.5, 140,
195 £10¡12 m2 . These give the mean value of squared
displacement for 30 seconds as 48.75 £ 10¡12 m2. Now,
from (7) and (35), we have
hx2(t)i = (kT t/3¼´a),
(36)
which on substituting the above listed values yields k =
1.36 £ 10¡23JK ¡1 . Clearly, it is quite close to the accepted value of the Boltzmann constant except that the
deviation is reasonably large.
It may be mentioned that the Langevin equation (equation (20)) was based on the assumptions enumerated in
the beginning of this section. However, in a real system
being used for the observation of Brownian motion, some
external force fext (t) may be present; the viscous drag
force f d may be a function of higher powers of velocity
or some other function of v(t); the drag coe±cient may
not be a constant and rather may depend upon v, t or
even position. In such a case this equation is modified
to a general form
M (dv/dt) = ¡ °(v, x, t)F (v) + f r (t) + fext (t).
(37)
It is worthwhile to point out that the Langevin equations can also be obtained by assuming the Brownian
particle to be interacting bilinearly with a large number of harmonic oscillators constituting a heat bath6 .
This approach yields explicit expressions for the drag
force f d as well as the stochastic force or noise fr (t) and,
thus, provides better insight into their mechanism but
still does not make the theory very rigorous7 . The latter
task is achieved by using the so called master equation
RESONANCE ⎜ August 2005
6
This d e riv a tion is g iv e n , e . g .,
in a rticl e 1.6 in Non e quilibrium
S t a t i st i c a l M e c h a n i cs b y R
Z w a n z i g ( O x f o r d U n iv e rsi ty
Press, Oxford, 2001).
Se e R Z w a nzig, J. S t a t. Phys .,
Vol.9, p .215, 1973.
7
65
GENERAL ⎜ ARTICLE
Se e H Riske n, Th e Fokker–
Planck Equation: Methods of S ol u t i o n a n d A p p l i c a t i o n s,
Spring er, 1996.
8
for the time rate of variation of probability distribution
function and its simplified version — the Fokker—Planck
equation.8 However, we do not propose to go into these
details here.
Suggested Reading
A d dress for Corresp on d e nce
Sh a m a Sh a rm a
De p a rtm e nt of Physics
D AV C oll e g e
B a thin d a 151 01
Punj a b, Indi a
Vishw a mittar
D e p a rtm e nt of Physics
P a nj a b University
Ch a n dig a rh 160 014, In di a .
Em a il:vm@puc. ac.in;
vishw a [email protected]
[1] E S R Gopal, Statistical Mechanics and Properties of Matter, Macmillan
India, Delhi, 1976.
[2] W Paul and J Baschnagel, Stochastic Processes from Physics to Finance,
Springer- Verlag, Berlin, 1999.
[3] R K Pathria, Statistical Mechanics, Butterworth-Heinemann, Oxford,
2001.
[4] J Klafter, M F Shlesinger and G Zumofen, Beyond Brownian Motion,
Physics Today, Vol. 49, No. 2, pp .33-39, 1996.
[5] A R Bulsara and L Gammaitoni, Tuning in to noise, Physics Today, Vol.
49, No. 3, pp. 39-45, 1996.
[6] M D Haw, Colloidal Suspensions, Brownian Motion, Molecular reality:
a Short History, J. Phys.: Condens. Matter, Vol. 14, pp .7769-79, 2002.
[7] R D Astumian and P Hanggi, Brownian Motors, Physics Today, Vol. 55,
No. 11,pp.33-39, 2002.
[8] D Ruelle, Conversations on Nonequilibrium Physics with an
Extraterrestial, Physics Today, Vol. 57, No. 5, pp.48-53, 2004.
Sir Isaac Newton had a theory of how to get the
best outcomes in a courtroom. He suggested to
lawyers that they should drag their arguments
into the late afternoon hours. The English judges
of his day would never abandon their 4 o’clock
tea time, and therefore would always bring
down their hammer and enter a hasty, positive
decision so they could retire to their chambers
for a cup of Earl Grey.
This tactic used by the British lawyers is still
recalled as ‘Newton’s Law of Gavel Tea’ in
legal circles.
66
RESONANCE ⎜ August 2005